Questions: Reductio ad Absurdum: Proof by Contradiction

Question 1

A student wants to prove there is no largest prime. She begins: 'Suppose there IS a largest prime p.' After reasoning, she shows a number larger than p must also be prime — contradicting her assumption. She concludes there is no largest prime. This is:

• A. Proof by induction, since she considers primes in sequence
• B. A direct proof, since she constructs a new prime explicitly
• C. Reductio ad absurdum — she assumes the negation of her conclusion and derives a contradiction
• D. A fallacy, because assuming a false statement can prove anything

Answer: (C)

She assumed the negation ('there IS a largest prime') and derived a contradiction (a prime larger than the supposed largest). The contradiction forces rejection of the assumption, establishing that there is no largest prime. Option D confuses 'assuming a false statement' with 'deriving a contradiction from it' — the whole point is that the contradiction reveals the assumption is false.

Question 2

Why is reductio especially useful when a direct proof is difficult?

• A. Because it lets you skip the burden of proof by showing an alternative
• B. Because it is often easier to show what goes wrong if the conclusion is false than to construct a direct argument for it
• C. Because reductio always produces shorter proofs than direct methods
• D. Because reductio allows you to use premises that haven't been proven yet

Answer: (B)

The method's power is asymmetric: sometimes the path from ¬P to contradiction is clearly visible even when the path from premises to P is not. By testing the negation, you can harness impossibility as a proof tool — showing that the world where ¬P holds is logically incoherent. This is exactly why the irrationality of √2 is so elegantly proved by reductio.

Question 3

A reductio ad absurdum proof is mainly valid if the contradiction it derives is a formal logical contradiction of the form 'P and not-P.'

• T. True
• F. False

Answer: False

The contradiction can be any statement known to be false — a known mathematical fact, an established premise, or any established truth — not just a tautological P ∧ ¬P. In the √2 proof, the contradiction is that a fraction assumed to be in lowest terms turns out to have a common factor of 2. This is logically sufficient to collapse the argument.

Question 4

Reductio ad absurdum can establish the truth of a statement by showing that its negation leads to a contradiction.

• T. True
• F. False

Answer: True

This is exactly the method's logical foundation. If ¬P → contradiction, and contradictions are necessarily false, then ¬P must be false, meaning P must be true. The inference relies on the law of excluded middle (either P or ¬P) and the principle that contradictions cannot hold — both standard commitments in classical logic.

Question 5

Explain why reductio ad absurdum is sometimes more powerful than direct proof, using the structure of the method as your explanation.

Model answer: Reductio works by testing the negation of what you want to prove. Sometimes the path from the negation to a contradiction is clearly visible — the assumption ¬P generates consequences that are obviously incompatible with known facts. A direct proof must construct a positive route from premises to conclusion, which may require insight that isn't available. Reductio converts impossibility into a proof tool: you don't need to know HOW the conclusion is true, only that its negation leads somewhere impossible.

The √2 irrationality proof illustrates this perfectly: there is no simple direct argument that √2 is irrational. But the assumption that it IS rational quickly unravels into a contradiction about odd and even numbers. The method is powerful precisely when the structure of what-goes-wrong is more visible than the structure of why-it's-true.